Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent nonzero real numbers. See Examples 5 and 6. y^8/y^12

Verified step by step guidance

1. Identify the expression to simplify: $\frac{y^8}{y^{12}}$.
2. Apply the quotient rule for exponents: $\frac{a^m}{a^n}=a^{m-n}$.
3. Subtract the exponents: $y^{8-12}=y^{-4}$.
4. Convert the negative exponent to a positive exponent: $y^{-4}=\frac{1}{y^4}$.
5. The simplified expression with positive exponents is $\frac{1}{y^4}$.

Key Concepts

Exponent Rules

Exponent rules are fundamental principles that govern the operations involving powers of numbers and variables. Key rules include the product of powers, quotient of powers, and power of a power. For instance, when dividing like bases, you subtract the exponents: a^m / a^n = a^(m-n). Understanding these rules is essential for simplifying expressions with exponents.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the opposite positive exponent. For example, a^(-n) = 1/a^n. In the context of simplification, it is important to express answers without negative exponents, which often involves rewriting the expression to ensure all terms are in positive exponent form. This helps maintain clarity and standard form in mathematical expressions.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing fractions to their simplest form by canceling common factors. In the case of exponents, this means applying the quotient rule to reduce the expression y^8/y^12. The goal is to express the result in a way that is easy to interpret and free of unnecessary complexity, which is particularly important in algebraic manipulations.